Calculus II PLTL Fall 2014 Worksheet 10 These problems are to be done without the use of a calculator unless otherwise specified 1 Pairs a Explain without formulas what a power series is b Explain without formulas what the interval of convergence of a power series is c List the six possible forms of the interval of convergence d Explain without formulas what the phrase radius of convergence of a power series means e How are power series different from series you have been working with in previous weeks 2 Scribe Find the interval and radius of convergence of the power series Be sure to check each endpoint of the interval for convergence P n 0 2x 1 n n2 1 3 Round Robin Let P5 x c0 c1 x c2 x2 c3 x3 c4x4 c5 x5 be the Taylor polynomial of degree 5 for the function f x x2 e 2x centered about the origin a How do you calculate the coefficients c0 c1 etc b What is P5 1 c What is the value of f 1 d The graph below shows both P5 x and f x Would you feel comfortable using it to approximate the value of f 0 05 for example What is the advantage of using P5 x instead of f x P 4 Pairs a Show that the series n 1 an with an 1 n2 converges using the Integral Test P b Suppose that s n 1 an Use the remainder estimate for the Pnintegral test to find the smallest integer n such that Rn s sn 0 01 where sn k 1 an 5 Round Robin For parts A D determine which of the following is true a Series I is convergent but the others are divergent b Series II is convergent but the others are divergent c Series III is convergent but the others are divergent d Series I and Series II are convergent but Series III is divergent e Series I and Series III are convergent but Series II is divergent f Series II and Series III are convergent but Series I is divergent g All three series are convergent h All three series are divergent i There is not enough information to determine the convergence divergence of each of the series A I P B I P C I D I P 1 n 1 5n II 1 n 2 n ln n 2 II P II P n 1 P 1 n 1 n2 1 P en n 1 nn n 1 n 1 n2 1 4n3 2n 1 III 2n 7n 5n III ln n n 1 n2 1 III n 1 Hint Compare ln n to n P II Hint n 1 sin 1 n limx 0 sin x x III 1 P n 1 P n 1 5 4 n 2 32n 1 1 n3 2 n 1 P 1 n 1 n2 3n P 1 n 1 3n n 1
View Full Document
Unlocking...